Physical Mathematics and the Future

The “vision” talk at Strings 2014 that I found most interesting was that of Greg Moore, whose topic was “Physical Mathematics and the Future”. He has a very extensive written version of the talk here, which includes both what he said, as well as a lot of detail about current topics at the interface of mathematics and physics.

I think what Moore has to say is quite fascinating, he’s giving a wonderful survey of where this intellectual subject is, how it has gotten there, and where it might be going. I’m very much in sympathy with his historical discussion of how math and physics have evolved, at times closely linked, at others finding themselves far apart. He’s concerned about an issue that I’ve commented on elsewhere, the fact that physics and math seem to be growing apart again, with no mathematicians speaking at the conference, instead attending their own conference (“Strings-Math 2014″). Physics departments increasingly want nothing to do with mathematics, which is a shame. One reason that Moore gave for this I found surprising, the idea that

most mathematicians are not as fully blessed with the opportunities for pursuing their research that many theoretical physicists enjoy.

It seems there’s a perception among many physicists that research mathematicians labor under some sort of difficult conditions of low pay and high teaching loads, but I think this is a misconception. Moore may be generalizing too much from the situation at Rutgers, where very unusual positions were created for string theorists at the height of that subject’s influence. From what I’ve seen, the salaries of top research mathematicians and theoretical physicists are quite comparable (if you don’t believe me, do some searches in the on-line data of salaries of faculty employed by public universities). Senior mathematicians do sometimes have slightly higher teaching loads, although often with a freedom to teach what they want. At the postdoc level, it is true that theoretical physics postdocs typically have no teaching, while similar positions in math often do require teaching. On the other hand, the job situation in theoretical physics is much more difficult than in mathematics. I’d say that working in an environment where you know you’re likely to find a permanent job is much preferable to one where you know this is unlikely, with doing some teaching not at all a significant problem.

On the question “What is String Theory?”, Moore’s take was that the “What is M-theory?” question is no longer getting much attention, with people kind of giving up. There was a very odd exchange at the end of the talk, when Witten asked him if he thought that maybe people should be emphasizing the string question, not the M-theory question, and Moore responded that the emphasis on M-theory was something he had learned from Witten himself.

His main point about this though was one I very much agree with, that the more interesting question now is “What is QFT?”. The standard way of thinking about QFTs in terms of an action principle doesn’t capture much of the interesting things about QFT we have learned over the years. Moore emphasizes certain examples, such as the (2,0) 6d superconformal theories, but discusses in his written version the relation of QFT to representation theory of some infinite dimensional groups, which I think provides even better examples of a different and more powerful way of thinking about QFT.

The written version contains a wealth of information surveying current topics in this area, is highly recommended to anyone who wants to try and understand what people working on “string theory” and mathematics have been up to. It appears that this document is a work in progress, with more material possibly to come (for instance, there’s a section 4.4 on Geometric Representation Theory still to be filled in). I look forward to future versions.

15 Responses to Physical Mathematics and the Future

All these questions about the meaning of QFT etc. originated and formulated in a meaningful way within String theory, so trying to decouple them is unnatural to say the least; they can be understood and addressed only within this general framework provided by String theory.

In the future the degree of convergence between the two will increase even more and I believe at some point there would be no distinction in practice between a Quantum field theorist and a String theorist.

I see that you’re using Strominger’s definition of “string theorist”.
The great thing about Moore’s talk was that it had a lot of serious ideas and content, wasn’t just the sort of “math shows that string theory rules” sloganeering that goes on in “vision” talks like this.

Towards the end he says “On the other hand, the relative lack of reliance of Physical Mathematics on laboratory experiments is viewed – with some justification – as dangerous by many physicists. The dangers of relying on “pure thought” when divining the secrets of Nature are well-known and illustrated by multitudinous examples.” Just so. The name Physical Mathematics is well chosen: virtually all the topics he discusses have have no clear relation to the physical properties of the real universe, so its mathematics rather than physics.
And that seems to apply also to the references to QFT in the paper (“Moore emphasizes certain examples, such as the (0,2) 6d superconformal theories”). Is there something useful there about the 4-d QFT that apparently describes the real universe? Or are these QFT theories physical mathematics rather then mathematical physics?

I think we already know the QFT that describes the real universe, it’s called the Standard Model (sure, it has a problem with quantizing gravitational degrees of freedom, but hundreds of physicists will tell you they know what to do about that…). To me, the question is whether we can understand that QFT better, in particular finding explanations of aspects of it that now look rather ad hoc. The conjecture that a better understanding of a QFT like the SM is going to require a deeper understanding of QFTs in general (or at least of a class of them) seems reasonable, and such a deeper understanding may very well involve new ideas about the relation of qfts to mathematics. That there is a vast amount we don’t understand about qfts and mathematics is undeniable. One can’t be sure that understanding more will help understand the real world, but one can be sure that it will lead to new mathematics. In any case, it seems to me more worth doing that most of what theorists working on fundamental physics spend their time on these days.

My own personal take on promising ways to go forward with this overlaps in some places with Moore’s, not at all in others.

Giotis: there is at least one area of mathematical physics which is asking very strongly and deeply what qft is, and it has nothing to do with string theory. This is called algebraic qft.
There is at least one other area of mathematical physics which is asking very much why the standard model is as it is, and it has nothing to do with string theory. This is called noncommutative geometry.
Is it asking string theorists too much to have a little bit of culture out of their field ?

Martibal,
I really don’t want to argue since the purpose of my comment wasn’t to claim that there is no QFT research outside String theory but to highlight the fact that the two are so intimately interrelated nowadays that one is forced to treat them within the context of one general, unified theoretical framework.

BTW saying that non commutative geometry has nothing to do with String theory is absurd. The very presence of D-branes renders space non commutative. Non commutative geometry is a hot topic in String theory research.

the MSSM is so intimately connected to the standard model that in fact it is equivalent to it in a certain, well-defined limit. it is fair to say that they are so much connected, that one is forced to treat both the SM and MSSM in the context of one general, unified theoretical framework

@ Chris – the noncommutative formulation of the standard model coupled to gravity in terms of spectral triples and an almost-commutative algebra does not permit MSSM (see http://arXiv.org/pdf/1211.0825.pdf). It is possible (but not natural) to incorporate supersymmetry in this framework, but there is certainly no equivalence.

in fact the spectral (Connes’s NCG-type) perspective on geometry, where Riemannian geometries are characterized by the quantum mechanics of superparticles roaming in them (that’s what a spectral triple encodes) is a simplified version of how effective target space geometry is encoded in string theory as whatever the quantum superstrings roaming in them sees, as witnessed by their worldsheet SCFT.

If I remember right, Connes’ original version of con-commutative standard model had a crisp prediction for the Higgs mass. A prediction that was immediately ruled out by Femilab, even before LHC.

Now one could argue whether Connes-like noncommutativity could be embedded in string theory. I am no expert on that, but my first gut reaction on hearing about it at the time, was: “no”. So I was happy that it was ruled out pretty instantly. It seemed way too much like a UV Lorentz-violating theory, but maybe I am forgetting.

@Urs: thank for the references. I did not know them so I will have a close look before commenting further. But at first sight I do not see where the “super particle roaming” in Connes description of the NCG are. Besides a Riemannian manifold and its Dirac operator, one has a finite dimensional algebra acting on a finite dimensional Hilbert space and a “finite dimensional Dirac operator” (in practical: a mass matrix). The fermions are the basis of this finite dimensional Hilbert space, the bosons (including the Higgs) are obtained as gauge fields (using a noncommutative generalization of a connection). I do not see super particles there. But I have to study your reference.

@Yatima and also: the prediction of the Higgs was indeed around 170 GeV (the uncertainty is mainly due to the uncertainty in the Top mass), and it has been ruled out by Tevratron in 2008. In Resilience of the Spectral Standard Model, Chamseddine and Connes have shown that by taking into account a new scalar field (proposed by particles physicist to cure the instability of the electroweak vacuum due to the “low” mass of the Higgs), then it was possible to get the correct mass for the Higgs. In the above mentioned paper, the way to introduce this new field following the “rules” of NCG was a bit rough. This has been made smoother in a successive paper of Chamseddine, Connes and v. Suijlekom [at this point I hope Peter will allow me to break the non-self-publicity rule to say that another way to generate the new field within the NCG framework had also been proposed by a “Napolitan team”]